課程名稱︰量子物理上 課程性質︰物理系系必修 課程教師︰陳智泓 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰01/10/12 考試時限（分鐘）：120min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.(10%) Compute the commutation relation, [e^ax, p], where a is a real constant. ^ 2. An operator A, representing observable A, has two normalized eigenstates Ψ1 and Ψ2, with eigenvalues a1 and a2 respectively. ^ Operator B, representing observable B, has two normalized eigenstates ψ1, ψ2, with eigenvalues b1 and b2. The eigenstates are related by: 1 1 Ψ1=---(4ψ1+3ψ2), Ψ2=---(3ψ1-4ψ2) 5 5 (a) (5%) Observable A is measured, and the value a1 is obtained. What is the state of the system (immediately) after this measurement? (b) (10%) If B is now measured, what are the possible results, and what are their probability? (c) (10%) Right after the measurement of B, A is measured again. What is the probability of getting a1? 3. The Hamiltonian for a certain three-level system is represented by matrix ^ ╭1 0 0╮ H= hw/2│0 1 0│ ╰0 0 2╯ Two other observalbes, A and B, are represented by the matrices ^ ╭0 1 0╮ A=λ│1 0 0│ ╰0 0 2╯ ^ ╭2 0 0╮ B=μ│0 0 1│ ╰0 1 0╯ where wψ,λ,μ are positive real numbers. ^ ^ ^ (a) (15%) Find the eigenvalues and (normalized) eigenvectors of H,A,B (b) (15%) Suppose the system starts out in the generic state ╭c1╮ |S(0)>=│c2│ ╰c3╯, ^ ^ ^ with |c1|^2 +|c2|^2 +|c2|^2 =1. Find the expextation value of H,A,B. (c) (25%) What is |S(t)>? If you measured the energy of this state (at time t), what values might you get, and what is the probability of each? Answer the same questions for A and B. 4. (10%) Quarks carry spin 1/2. A quark and its antiparticle bind together to make a meson. Assume the orbital angular momentum in a meson between the quark and the anti-quark is l=1. What angular momentum states are possible for meson? 5 (0%) 祝大家新年愉快，寒假過得充時開心！ 註:第五題不是我自己打的...老師是好人。 --

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